Potential Operators¶
Once the solution of a boundary integral formulation has been approximated, potential operators can be used to compute point evaluations of the solution inside the domain.
Full documentation of Bempp potential operators can be found on Read the Docs.
Potential Operators for Laplace’s Equation¶
For Laplace’s equation, there are two potential operators that are used, as given in the table below.
Operator |
Definition |
|---|---|
Single layer |
\((\mathcal{V}\mu)(\mathbf{x}) := \int_{\Gamma} G(\mathbf{x},\mathbf{y}) \mu(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
Double layer |
\((\mathcal{K}v)(\mathbf{x}) := \int_{\Gamma} \frac{\partial G(\mathbf{x},\mathbf{y})}{\partial\mathbf{\nu}_{\mathbf{y}}} v(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
In each case, \(G(\mathbf{x},\mathbf{y})\) is the Green’s function for Laplace’s equation.
To assemble potential operators in Bempp, the desired evaluation points must first be defined. For example, the following snippet creates a grid of 2500 points in the \(x\)\(y\)-plane with \(x\) and \(y\) between -3 and 3.
plot_grid = np.mgrid[-3:3:50j, -3:3:50j]
points = np.vstack((plot_grid[0].ravel(),
plot_grid[1].ravel(),
np.zeros(plot_grid[0].size)))
Potential operators can thenbe initialised in Bempp using:
from bempp.api.operators.potential import laplace
single = laplace.single_layer(domain, points)
double = laplace.double_layer(domain, points)
These can be applied to a grid function with:
single.evaluate(solution)
double.evaluate(solution)
As with boundary operators, assembler and device_interface
keyword arguments can be used to control the assembly type used for potential operators.
Potential Operators for the Helmholtz Equation¶
For the Helmholtz equation, there are two potential operators that are used, as given in the table below.
Operator |
Definition |
|---|---|
Single layer |
\((\mathcal{V}\mu)(\mathbf{x}) := \int_{\Gamma} G_k(\mathbf{x},\mathbf{y}) \mu(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
Double layer |
\((\mathcal{K}v)(\mathbf{x}) := \int_{\Gamma} \frac{\partial G_k(\mathbf{x},\mathbf{y})}{\partial\mathbf{\nu}_{\mathbf{y}}} v(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
In each case, \(G_k(\mathbf{x},\mathbf{y})\) is the Green’s function for the Helmholtz equation with wavenumber \(k\).
To assemble potential operators in Bempp, the desired evaluation points must first be defined. For example, the following snippet creates a grid of 2500 points in the \(x\)\(y\)-plane with \(x\) and \(y\) between -3 and 3.
plot_grid = np.mgrid[-3:3:50j, -3:3:50j]
points = np.vstack((plot_grid[0].ravel(),
plot_grid[1].ravel(),
np.zeros(plot_grid[0].size)))
Potential operators can thenbe initialised in Bempp using:
from bempp.api.operators.potential import helmholtz
single = helmholtz.single_layer(domain, points, wavenumber)
double = helmholtz.double_layer(domain, points, wavenumber)
These can be applied to a grid function with:
single.evaluate(solution)
double.evaluate(solution)
As with boundary operators, assembler and device_interface
keyword arguments can be used to control the assembly type used for potential operators.
Potential Operators for the modified Helmholtz Equation¶
For the modified Helmholtz equation, there are two potential operators that are used, as given in the table below.
Operator |
Definition |
|---|---|
Single layer |
\((\mathcal{V}\mu)(\mathbf{x}) := \int_{\Gamma} G_\omega(\mathbf{x},\mathbf{y}) \mu(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
Double layer |
\((\mathcal{K}v)(\mathbf{x}) := \int_{\Gamma} \frac{\partial G_\omega(\mathbf{x},\mathbf{y})}{\partial\mathbf{\nu}_{\mathbf{y}}} v(\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
In each case, \(G_\omega(\mathbf{x},\mathbf{y})\) is the Green’s function for the modified Helmholtz equation with frequency \(\omega\).
To assemble potential operators in Bempp, the desired evaluation points must first be defined. For example, the following snippet creates a grid of 2500 points in the \(x\)\(y\)-plane with \(x\) and \(y\) between -3 and 3.
plot_grid = np.mgrid[-3:3:50j, -3:3:50j]
points = np.vstack((plot_grid[0].ravel(),
plot_grid[1].ravel(),
np.zeros(plot_grid[0].size)))
Potential operators can thenbe initialised in Bempp using:
from bempp.api.operators.potential import modified_helmholtz
single = modified_helmholtz.single_layer(domain, points, omega)
double = modified_helmholtz.double_layer(domain, points, omega)
These can be applied to a grid function with:
single.evaluate(solution)
double.evaluate(solution)
As with boundary operators, assembler and device_interface
keyword arguments can be used to control the assembly type used for potential operators.
Potential Operators for Maxwell’s Equations¶
For Maxwell’s equations, there are two potential operators that are used, as given in the table below.
Operator |
Definition |
|---|---|
Electric field |
\((\mathcal{E}(\mathbf{p}))(\mathbf{x})=\mathrm{i} k\int_\Gamma\mathbf{p}(\mathbf{y})G_k(\mathbf{x},\mathbf{y})\,\mathrm{d}\mathbf{y}-\frac1{\mathrm{i} k}\nabla_{\mathbf{x}}\int_\Gamma\nabla_{\Gamma}\cdot\mathbf{p}(\mathbf{y})G_k(\mathbf{x},\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
Magnetic field |
\((\mathcal{H}(\mathbf{p})(\mathbf{x})=\nabla_\mathbf{x}\times\int_\Gamma\mathbf{p}(\mathbf{y})G(\mathbf{x},\mathbf{y})\,\mathrm{d}\mathbf{y}\) |
In each case, \(G_k(\mathbf{x},\mathbf{y})\) is the Green’s function for the Helmholtz equation with wavenumber \(k\).
To assemble potential operators in Bempp, the desired evaluation points must first be defined. For example, the following snippet creates a grid of 2500 points in the \(x\)\(y\)-plane with \(x\) and \(y\) between -3 and 3.
plot_grid = np.mgrid[-3:3:50j, -3:3:50j]
points = np.vstack((plot_grid[0].ravel(),
plot_grid[1].ravel(),
np.zeros(plot_grid[0].size)))
Potential operators can thenbe initialised in Bempp using:
from bempp.api.operators.potential import maxwell
electric = maxwell.electric_feild(domain, points, wavenumber)
magnetic = maxwell.magnetic_feild(domain, points, wavenumber)
These can be applied to a grid function with:
electric.evaluate(solution)
magnetic.evaluate(solution)
As with boundary operators, assembler and device_interface
keyword arguments can be used to control the assembly type used for potential operators.